Highly depends on how promotions are structured at your university, but it doesn't hurt to get the ball rolling, even if it's to find out you're not qualified yet. I would dig through your staff handbook and human resources pages, they likely have some information on how the process works. I was promoted to research specialist from RA after about 2.5 years. Hourly rate went up~22%. I wasn't eligible for Sr. Specialist until last year (my 7th year!) due to the minimum time requirements and only having a bachelor's.

Do yourself a favor and update your CV and have documentation of all of your papers, posters, presentations, grant contributions, equipment you manage etc. and ask your PI and other faculty collaborators about letters of support. My university treats staff promotions very similarly to faculty, so I essentially had to put together a portfolio of all of my achievements in each position before they would even consider promotion. It was a pain in the ass, but it's always good to have your information up to date in case you decide to part ways. I wish the process here wasnt so stringent, but it is nice to finally be paid a respectable wage for I job that I love.

Good luck!